Fixed-Time Convergence for a Class of Nonconvex-Nonconcave Min-Max Problems

TL;DR

This paper introduces a fixed-time convergent saddle point dynamical system for min-max problems that relaxes convexity-concavity assumptions, achieving rapid convergence under the two-sided Polyak-Łojasiewicz inequality.

Contribution

It presents a novel dynamical system approach that guarantees fixed-time convergence for nonconvex-nonconcave min-max problems under the two-sided PL condition, broadening applicability.

Findings

Achieves arbitrarily fast convergence compared to existing methods.

Guarantees fixed-time convergence under weaker conditions.

Numerical results validate superior performance.

Abstract

This study develops a fixed-time convergent saddle point dynamical system for solving min-max problems under a relaxation of standard convexity-concavity assumption. In particular, it is shown that by leveraging the dynamical systems viewpoint of an optimization algorithm, accelerated convergence to a saddle point can be obtained. Instead of requiring the objective function to be strongly-convex--strongly-concave (as necessitated for accelerated convergence of several saddle-point algorithms), uniform fixed-time convergence is guaranteed for functions satisfying only the two-sided Polyak-{\L}ojasiewicz (PL) inequality. A large number of practical problems, including the robust least squares estimation, are known to satisfy the two-sided PL inequality. The proposed method achieves arbitrarily fast convergence compared to any other state-of-the-art method with linear or even super-linear convergence, as also corroborated in numerical case studies.